EGO with a failure region

import numpy as np
import tensorflow as tf

np.random.seed(1234)
tf.random.set_seed(1234)

The problem

This notebook is similar to the EI notebook, where we look to find the minimum value of the two-dimensional Branin function over the hypercube \([0, 1]^2\). But here, we constrain the problem, by adding an area to the search space in which the objective fails to evaluate.

We represent this setup with a function masked_branin that produces null values when evaluated in the disk with center \((0.5, 0.4)\) and radius \(0.3\). It’s important to remember that while we know where this failure region is, this function is a black box from the optimizer’s point of view: the optimizer must learn it.

import trieste

def masked_branin(x):
    mask_nan = np.sqrt((x[:, 0] - 0.5) ** 2 + (x[:, 1] - .4) ** 2) < 0.3
    y = np.array(trieste.utils.objectives.branin(x))
    y[mask_nan] = np.nan
    return tf.convert_to_tensor(y.reshape(-1, 1), x.dtype)

As mentioned, we’ll search over the hypercube \([0, 1]^2\) …

search_space = trieste.space.Box([0, 0], [1, 1])

… where the masked_branin now looks as follows. The white area in the centre shows the failure region.

from util.plotting_plotly import plot_function_plotly

fig = plot_function_plotly(
    masked_branin, search_space.lower, search_space.upper, grid_density=70
)
fig.update_layout(height=400, width=400)
fig.show()

Define the data sets

We’ll work with two data sets

• one containing only those query_points and observations where the observations are finite. We’ll label this with OBJECTIVE.

• the other containing all the query points, but whose observations indicate if evaluating the observer failed at that point, using 1 if the evaluation failed, else 0. We’ll label this with FAILURE.

Let’s define an observer that outputs the data in these formats.

OBJECTIVE = "OBJECTIVE"
FAILURE = "FAILURE"

def observer(x):
    y = masked_branin(x)
    mask = np.isfinite(y).reshape(-1)
    return {
        OBJECTIVE: trieste.data.Dataset(x[mask], y[mask]),
        FAILURE: trieste.data.Dataset(x, tf.cast(np.isfinite(y), tf.float64))
    }

We can evaluate the observer at points sampled from the search space.

num_init_points = 15
initial_data = observer(search_space.sample(num_init_points))

Model the data

We’ll model the data on the objective with a regression model, and the data on which points failed with a classification model. The regression model will be a GaussianProcessRegression wrapping a GPflow GPR, and the classification model a VariationalGaussianProcess wrapping a GPflow VGP with Bernoulli likelihood.

import gpflow


def create_regression_model(data):
    variance = tf.math.reduce_variance(data.observations)
    kernel = gpflow.kernels.Matern52(variance=variance, lengthscales=[0.2, 0.2])
    gpr = gpflow.models.GPR(data.astuple(), kernel, noise_variance=1e-5)
    gpflow.set_trainable(gpr.likelihood, False)
    return gpr


def create_classification_model(data):
    kernel = gpflow.kernels.SquaredExponential(
        variance=100.0, lengthscales=[0.2, 0.2]
    )
    likelihood = gpflow.likelihoods.Bernoulli()
    vgp = gpflow.models.VGP(data.astuple(), kernel, likelihood)
    gpflow.set_trainable(vgp.kernel.variance, False)
    return vgp


regression_model = create_regression_model(initial_data[OBJECTIVE])
classification_model = create_classification_model(initial_data[FAILURE])

Create a custom optimize method

The new NatGradTrainedVGP class has a custom optimize method that alternates between Adam steps to optimize the lengthscales and NatGrad steps to optimize the variational parameters:

class NatGradTrainedVGP(trieste.models.VariationalGaussianProcess):
    def optimize(self, dataset):
        gpflow.set_trainable(self.model.q_mu, False)
        gpflow.set_trainable(self.model.q_sqrt, False)
        variational_params = [(self.model.q_mu, self.model.q_sqrt)]
        adam_opt = tf.optimizers.Adam(1e-3)
        natgrad_opt = gpflow.optimizers.NaturalGradient(gamma=0.1)

        for step in range(50):
            loss = self.model.training_loss
            natgrad_opt.minimize(loss, variational_params)
            adam_opt.minimize(loss, self.model.trainable_variables)

We’ll train the GPR model with an L-BFGS-based optimizer, and the GPC model with the custom algorithm above.

from typing import Dict

models: Dict[str, trieste.models.ModelSpec] = {
    OBJECTIVE: {
        "model": regression_model,
        "optimizer": gpflow.optimizers.Scipy(),
        "optimizer_args": {
            "minimize_args": {"options": dict(maxiter=100)},
        },
    },
    FAILURE: NatGradTrainedVGP(classification_model),
}

Create a custom acquisition function

We’ll need a custom acquisition function for this problem. This function is the product of the expected improvement for the objective data and the predictive mean for the failure data. We can specify which data and model to use in each acquisition function builder with the OBJECTIVE and FAILURE labels. We’ll optimize the function using EfficientGlobalOptimization.

class ProbabilityOfValidity(trieste.acquisition.SingleModelAcquisitionBuilder):
    def prepare_acquisition_function(self, dataset, model):
        return lambda at: trieste.acquisition.lower_confidence_bound(model, 0.0, at)

ei = trieste.acquisition.ExpectedImprovement()
pov = ProbabilityOfValidity()
acq_fn = trieste.acquisition.Product(ei.using(OBJECTIVE), pov.using(FAILURE))
rule = trieste.acquisition.rule.EfficientGlobalOptimization(acq_fn)

Run the optimizer

Now, we run the Bayesian optimization loop for twenty steps, and print the location of the query point corresponding to the minimum observation.

bo = trieste.bayesian_optimizer.BayesianOptimizer(observer, search_space)

result = bo.optimize(20, initial_data, models, rule).final_result.unwrap()

arg_min_idx = tf.squeeze(tf.argmin(result.datasets[OBJECTIVE].observations, axis=0))
print(f"query point: {result.datasets[OBJECTIVE].query_points[arg_min_idx, :]}")

Optimization completed without errors
query point: [0.12224536 0.81236841]

We can visualise where the optimizer queried on a contour plot of the Branin with the failure region. The minimum observation can be seen along the bottom axis towards the right, outside of the failure region.

import matplotlib.pyplot as plt
from util.plotting import plot_gp_2d, plot_function_2d, plot_bo_points

mask_fail = result.datasets[FAILURE].observations.numpy().flatten().astype(int) == 0
fig, ax = plot_function_2d(
    masked_branin,
    search_space.lower,
    search_space.upper,
    grid_density=50,
    contour=True
)
plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    ax=ax[0, 0],
    num_init=num_init_points,
    mask_fail=mask_fail,
)
plt.show()

We can also plot the mean and variance of the predictive distribution over the search space, first for the objective data and model …

from util.plotting_plotly import plot_gp_plotly, add_bo_points_plotly

arg_min_idx = tf.squeeze(tf.argmin(result.datasets[OBJECTIVE].observations, axis=0))

fig = plot_gp_plotly(
    result.models[OBJECTIVE].model,
    search_space.lower,
    search_space.upper,
    grid_density=50
)
fig = add_bo_points_plotly(
    x=result.datasets[OBJECTIVE].query_points[:, 0].numpy(),
    y=result.datasets[OBJECTIVE].query_points[:, 1].numpy(),
    z=result.datasets[OBJECTIVE].observations.numpy().flatten(),
    num_init=num_init_points,
    idx_best=arg_min_idx,
    fig=fig,
    figrow=1,
    figcol=1,
)

fig.show()

… and then for the failure data and model

fig, ax = plot_gp_2d(
    result.models[FAILURE].model,
    search_space.lower,
    search_space.upper,
    grid_density=50,
    contour=True,
    figsize=(12, 5),
    predict_y=True,
)

plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    num_init=num_init_points,
    ax=ax[0, 0],
    mask_fail=mask_fail,
)

plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    num_init=num_init_points,
    ax=ax[0, 1],
    mask_fail=mask_fail,
)

plt.show()

LICENSE

Apache License 2.0